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Goryachev-Chaplygin top

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A rigid body rotating about a fixed point, for which:

a) the principal moments of inertia , with regard to the fixed point, satisfy the relation ;

b) the centre of mass belongs to the equatorial plane through the fixed point;

c) the principal angular momentum is perpendicular to the direction of gravity, i.e., . First introduced by D. Goryachev [a4] in 1900, the system was later integrated by S.A. Chaplygin [a3] in terms of hyper-elliptic integrals (cf. also Hyper-elliptic integral). The system merely satisfying a) and b) is not algebraically integrable, but on the locus, defined by c), it is; namely, it has an extra invariant of homogeneous degree :

C. Bechlivanidis and P. van Moerbeke [a1] have shown that the problem has asymptotic solutions which are meromorphic in ; the system linearizes on a double cover of a hyper-elliptic Jacobian (i.e., of the Jacobi variety of a hyper-elliptic curve; cf. also Plane real algebraic curve), ramified exactly along the two hyper-elliptic curves, where the phase variables blow up; see also [a5]. An elementary algebraic mapping transforms the Goryachev–Chaplygin equations into equations closely related to the -body Toda lattice. A Lax pair is given in [a2]:

where and are given by the right-lower corner of and and where

See also Kowalewski top.

References

[a1] C. Bechlivanidis, P. van Moerbeke, "The Goryachev–Chaplygin top and the Toda lattice" Comm. Math. Phys. , 110 (1987) pp. 317–324
[a2] A.I. Bobenko, V.B. Kuznetsov, "Lax representation and new formulae for the Goryachev–Chaplygin top" J. Phys. A , 21 (1988) pp. 1999–2006
[a3] S.A. Chaplygin, "A new case of rotation of a rigid body, supported at one point" , Collected works , I , Gostekhizdat (1948) pp. 118–124 (In Russian)
[a4] D. Goryachev, "On the motion of a rigid material body about a fixed point in the case " Mat. Sb. , 21 (1900) (In Russian)
[a5] L. Piovan, "Cyclic coverings of Abelian varieties and the Goryachev–Chaplygin top" Math. Ann. , 294 (1992) pp. 755–764
How to Cite This Entry:
Goryachev-Chaplygin top. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Goryachev-Chaplygin_top&oldid=22523
This article was adapted from an original article by P. van Moerbeke (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article